I am interested in knowing whether the some of the European politicians' claims against the provision of shelter to refugees, asylum seekers, and - for lack of a better term - fortune seekers from the MENA region, hold any historical merit. For example, it is debated that the large influx of Germanic tribes has played a part in the downfall of the Roman Empire, yet this could be blamed on Rome expanding too quickly for its own good.
Have there been any examples in history that back up the politicians' claims?
While I can't answer your question coherently, I can link to a debunking of the "immigration caused the downfall of Rome" myth. https://www.reddit.com/r/badhistory/comments/3dzi9i/horse_is_lead_to_water_refuses_to_drink_proclaims/
Well about the Rome thing, they had been done expanding for a very long time by the time all those Germans showed up.